Crucial to our updating scheme is an appropriate choice of variables. We pursue a conservative approach in which \(\rho _{\sigma }\)  and \(\Phi _{\sigma }\)  are packaged together as canonically conjugate variables following the prescription detailed in IAC [1]}. The nontrivial task of choosing the geometric variables has long been the subject of a lively debate.Just to name a few approaches, there are, of course, the original attempts at geometrodynamics from Dirac [2]} as well as Arnowitt, Deser, and Misner (ADM) [3]} that start from the Einstein-Hilbert action and take the metric \(g_{ij}\)  as the fundamental building block. Somewhat more recently, due in part to the modern success of gauge theories, there has been some interest in taking, not the metric \(g_{ij}\) , but the Levi-Civita connection \(\Gamma ^{i}_{jk}\) , as the fundamental gravitational object (see e.g., [4]}). In a similar spirit is the well-known discovery of Ashtekar [5]}, which uses triads and spin connections to rewrite GR. Among these various approaches, however, the pioneering efforts of Hojman, Kuchař, and Teitelboim (HKT) [6]} prove to be of special interest, as their work centers a purely canonical approach to the dynamics of geometry. Following HKT, it is possible to take the six components of the metric \(g_{ij}\)  to be the starting point for a geometrodynamics. The argument for this is grounded in simplicity. If one assumes, as they do, that evolution in local time should mirror the structure of surface deformations, then the metric appearing on the right hand side of () for the “structure constant" \(\kappa ^{i}_{\perp \perp }\)  must necessarily be a functional of the canonical variables. Clearly this is most easily satisfied if the metric is itself a canonical variable.
